14 research outputs found
Linear Time Parameterized Algorithms via Skew-Symmetric Multicuts
A skew-symmetric graph is a directed graph with an
involution on the set of vertices and arcs. In this paper, we
introduce a separation problem, -Skew-Symmetric Multicut, where we are given
a skew-symmetric graph , a family of of -sized subsets of
vertices and an integer . The objective is to decide if there is a set
of arcs such that every set in the family has a vertex
such that and are in different connected components of
. In this paper, we give an algorithm for
this problem which runs in time , where is the
number of arcs in the graph, the number of vertices and the length
of the family given in the input.
Using our algorithm, we show that Almost 2-SAT has an algorithm with running
time and we obtain algorithms for {\sc Odd Cycle Transversal}
and {\sc Edge Bipartization} which run in time and
respectively. This resolves an open problem posed by Reed,
Smith and Vetta [Operations Research Letters, 2003] and improves upon the
earlier almost linear time algorithm of Kawarabayashi and Reed [SODA, 2010].
We also show that Deletion q-Horn Backdoor Set Detection is a special case of
3-Skew-Symmetric Multicut, giving us an algorithm for Deletion q-Horn Backdoor
Set Detection which runs in time . This gives the first
fixed-parameter tractable algorithm for this problem answering a question posed
in a paper by a superset of the authors [STACS, 2013]. Using this result, we
get an algorithm for Satisfiability which runs in time where
is the size of the smallest q-Horn deletion backdoor set, with being
the length of the input formula
Representative set statements for delta-matroids and the Mader delta-matroid
We present representative sets-style statements for linear delta-matroids,
which are set systems that generalize matroids, with important connections to
matching theory and graph embeddings. Furthermore, our proof uses a new
approach of sieving polynomial families, which generalizes the linear algebra
approach of the representative sets lemma to a setting of bounded-degree
polynomials. The representative sets statements for linear delta-matroids then
follow by analyzing the Pfaffian of the skew-symmetric matrix representing the
delta-matroid. Applying the same framework to the determinant instead of the
Pfaffian recovers the representative sets lemma for linear matroids.
Altogether, this significantly extends the toolbox available for kernelization.
As an application, we show an exact sparsification result for Mader networks:
Let be a graph and a partition of a set of terminals , . A -path in is a path with endpoints
in distinct parts of and internal vertices disjoint from . In
polynomial time, we can derive a graph with ,
such that for every subset there is a packing of
-paths with endpoints in if and only if there is one in
, and . This generalizes the (undirected version of the)
cut-covering lemma, which corresponds to the case that contains
only two blocks.
To prove the Mader network sparsification result, we furthermore define the
class of Mader delta-matroids, and show that they have linear representations.
This should be of independent interest
Covering Small Independent Sets and Separators with Applications to Parameterized Algorithms
We present two new combinatorial tools for the design of parameterized
algorithms. The first is a simple linear time randomized algorithm that given
as input a -degenerate graph and an integer , outputs an independent
set , such that for every independent set in of size at most ,
the probability that is a subset of is at least .The second is a new (deterministic) polynomial
time graph sparsification procedure that given a graph , a set of terminal pairs and an
integer , returns an induced subgraph of that maintains all
the inclusion minimal multicuts of of size at most , and does not
contain any -vertex connected set of size . In
particular, excludes a clique of size as a
topological minor. Put together, our new tools yield new randomized fixed
parameter tractable (FPT) algorithms for Stable - Separator, Stable Odd
Cycle Transversal and Stable Multicut on general graphs, and for Stable
Directed Feedback Vertex Set on -degenerate graphs, resolving two problems
left open by Marx et al. [ACM Transactions on Algorithms, 2013]. All of our
algorithms can be derandomized at the cost of a small overhead in the running
time.Comment: 35 page
Parameterized complexity and approximability of directed odd cycle transversal
A directed odd cycle transversal of a directed graph (digraph) D is a vertex set S that intersects every odd directed cycle of D. In the Directed Odd Cycle Transversal (DOCT) problem, the input consists of a digraph D and an integer k. The objective is to determine whether there exists a directed odd cycle transversal of D of size at most k. In this paper, we settle the parameterized complexity of DOCT when parameterized by the solution size k by showing that DOCT does not admit an algorithm with running time f(k)nO(1) unless FPT = W[1]. On the positive side, we give a factor 2 fixed-parameter approximation (FPT approximation) algorithm for the problem. More precisely our algorithm takes as input D and k, runs in time 2O(k)nO(1), and either concludes that D does not have a directed odd cycle transversal of size at most k, or produces a solution of size at most 2k. Finally assuming gap-ETH, we show that there exists an ϵ > 0 such that DOCT does not admit a factor (1 + ϵ) FPT-approximation algorithm
Linear-Time FPT Algorithms via Network Flow
In the area of parameterized complexity, to cope with NP-Hard problems, we
introduce a parameter k besides the input size n, and we aim to design
algorithms (called FPT algorithms) that run in O(f(k)n^d) time for some
function f(k) and constant d. Though FPT algorithms have been successfully
designed for many problems, typically they are not sufficiently fast because of
huge f(k) and d. In this paper, we give FPT algorithms with small f(k) and d
for many important problems including Odd Cycle Transversal and Almost 2-SAT.
More specifically, we can choose f(k) as a single exponential (4^k) and d as
one, that is, linear in the input size. To the best of our knowledge, our
algorithms achieve linear time complexity for the first time for these
problems. To obtain our algorithms for these problems, we consider a large
class of integer programs, called BIP2. Then we show that, in linear time, we
can reduce BIP2 to Vertex Cover Above LP preserving the parameter k, and we can
compute an optimal LP solution for Vertex Cover Above LP using network flow.
Then, we perform an exhaustive search by fixing half-integral values in the
optimal LP solution for Vertex Cover Above LP. A bottleneck here is that we
need to recompute an LP optimal solution after branching. To address this
issue, we exploit network flow to update the optimal LP solution in linear
time.Comment: 20 page
Solving hard cut problems via flow-augmentation
We present a new technique for designing FPT algorithms for graph cut
problems in undirected graphs, which we call flow augmentation. Our technique
is applicable to problems that can be phrased as a search for an (edge)
-cut of cardinality at most in an undirected graph with
designated terminals and .
More precisely, we consider problems where an (unknown) solution is a set of size at most such that (1) in , and are in
distinct connected components, (2) every edge of connects two distinct
connected components of , and (3) if we define the set as these edges for which there exists an -path with
, then separates from . We prove that
in this scenario one can in randomized time add a
number of edges to the graph so that with probability no
added edge connects two components of and becomes a minimum cut
between and .
We apply our method to obtain a randomized FPT algorithm for a notorious
"hard nut" graph cut problem we call Coupled Min-Cut. This problem emerges out
of the study of FPT algorithms for Min CSP problems, and was unamenable to
other techniques for parameterized algorithms in graph cut problems, such as
Randomized Contractions, Treewidth Reduction or Shadow Removal.
To demonstrate the power of the approach, we consider more generally Min
SAT(), parameterized by the solution cost. We show that every problem
Min SAT() is either (1) FPT, (2) W[1]-hard, or (3) able to express the
soft constraint , and thereby also the min-cut problem in directed
graphs. All the W[1]-hard cases were known or immediate, and the main new
result is an FPT algorithm for a generalization of Coupled Min-Cut